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1. Introduction
Consider the following lattice differential equation
(1.1)
where
,
are positive constants,
,
is a
-function, and
.
Lattice dynamical systems occur in a wide variety of applications, and a lot of studies have been done, e.g., see [1]-[4]. A pair of solutions
,
of (1.1) is called a traveling wave solution with wave
speed
if there exist functions
such that
,
with
and
. Let
, note that (1.1) has a pair of traveling wave solutions if and only if
,
satisfy the functional differential equation
(1.2)
Without loss of generality, we can impose (1.1) with asymptotic boundary conditions
,
,
,
. (1.3)
By the property of equation, we can assume that
. In the following, we give some assumptions on nonlinear function
:
![]()
,
,
.
There exists a positive-value continuous function
such that
,
.
![]()
,
,
![]()
for any
,
,
where
,
is given in Lemma 2.1.
![]()
for any
.
Select positive constants
such that
,
, and define operators
by
![]()
. (1.4)
Then, (1.2) can be rewritten as
,
. (1.5)
Define the operators
by
.
Note that
satisfy
and a fixed point of
is a solution of (1.2). Denote
the Euclidean norm in
. Define
,
where
. Note that
is a Banach space.
Definition 1.1. If the continuous functions
are differentiable almost everywhere and satisfy
(1.6)
Then,
is called an upper solution of (1.2).
Similarity, we can define a lower solution of (1.2). The main result of this paper is
Theorem 1.1. Assume that
hold. Then there exists
such that for every
, (1.2) admits a traveling wave solution
connecting
and
. Moreover, each component of traveling wave solution is monotonically nondecreasing in
, and for each
,
,
also
satisfy
,
, where
is the smallest solution of the eq-
uation
.
2. Upper-Lower Solutions of (1.2)
Set
.
Lemma 2.1. Assume that
holds. Then there exists a unique
such that
if
, then there exist two positive numbers
and
with
such that
,
in
, and
in
;
if
, then
for all
;
if
, then
, and
.
Proof. Using assumption
, we can get the result directly. ![]()
Lemma 2.2. Assume that
,
and
hold. Let
,
, and
be defined as in
Lemma 2.1, and
be any number. Then for every
and
, there exists
such that for any
,
,
and
![]()
are a pair of upper solutions and a pair of lower solutions of (1.2), respectively.
Proof. Let
(2.1)
. (2.2)
Since
, there exists
such that
,
. If
, then
,
. By
, we get that
,
.
If
, then
. By
,
, and using
Lemma 2.1, we get that
![]()
(2.3)
Lemma 2.1 and
yields
. (2.4)
Thus,
![]()
Therefore,
is an upper solution of (1.2). Similarly, we can prove that
is a lower solution. ![]()
3. Existence of Traveling Wave
Let
,
. We have the fol-
lowing result.
Lemma 3.1 Assume that
and
hold. Then
![]()
and
for
if
satisfy
,
for
;
![]()
are nondecreasing in
if
is nondecreasing in
.
Proof. If
such that
and
for
, then by
we have
(3.1)
where
. Note that
(3.2)
Thus, from (3.1)-(3.2), we have
![]()
which implies that
. A similar argument can be done for
. Thus, we can get the desired results. ![]()
Lemma 3.2. Assume that
and
hold. Then
is continuous
with respect to the norm
with
.
Proof. We first prove that
are continuous. Denote ![]()
. For any
, choose
, where
. If
and
satisfy
, then by (3.1),
(3.3)
Similarly,
is continuous.
By definition of
, we have
(3.4)
If
, it follows that
. (3.5)
If
, it follows that
(3.6)
Combining (3.5) and (3.6), we get that
is continuous with respect to the norm
. A Similar argument can be done for
. ![]()
Define
![]()
It is easy to verify that
is nonempty, convex and compact in
. As the proof of Claim 2 in the proof of Theorem A in [5], we have
Lemma 3.3. Assume that
hold. Then
.
Proof of Theorem 1.1. By the definition of
, Lemma 3.2-3.3 and Schauder’s fixed point theorem, we get that there exists a fixed point
. Note that
is nondecreasing in
, as-
sumption
and Lemma 2.2 imply that
. There-
fore,
is a traveling wave solution of (1.1). ![]()
Acknowledgements
This work was supported by the NNSF of China Grant 11571092.